In summary, Theorem REF  tells us that when the coefficients \(b,f\)  and \(g\)  of the game are sufficiently regular, the Hamiltonian satisfies a certain convexity condition and the master equation corresponding to the mean field game admits a Lipschitz–continuous solution, then, not only that any sequence of Nash equilibria converges to a mean field equilibrium, but in addition the sequence of Nash equilibria satisfies the LDP.
The condition pertaining to existence and Lipschitz–continuity of the solution of the master equation (REF ) (i.e. REF ) is the only hard-to-check condition we impose here.
That being said, conditions guarantying existence and uniqueness of the master equation are given by [1]} and [2]}.
These authors study for instance the equation arising from a mean field game (albeit without interaction through the controls) in [1]}.
In section REF  below we discuss an example for which the equation is known (from the work [1]}) to have a Lipschitz–continuous solution.
Let us also refer to [5]}, [6]}, [7]}, [8]} for more recent results on the existence of the master equation.
